Lattices Appendix A Computer Security: Art and Science, 2 nd Edition - PowerPoint PPT Presentation

Lattices Appendix A Computer Security: Art and Science, 2 nd Edition Version 1.0 Slide A-1

Outline • Overview • Definitions • Lattices • Examples Computer Security: Art and Science, 2 nd Edition Version 1.0 Slide A-2

Overview • Lattices used to analyze several models • Bell-LaPadula confidentiality model • Biba integrity model • A lattice consists of a set and a relation • Relation must partially order set • Relation orders some, but not all, elements of set Computer Security: Art and Science, 2 nd Edition Version 1.0 Slide A-3

Sets and Relations • S set, R : S ´ S relation • If a , b Î S , and ( a , b ) Î R , write aRb • Example • I = { 1, 2, 3 }; R is ≤ • R = { (1, 1), (1, 2), (1, 3), (2, 2), (2, 3), (3, 3) } • So we write 1 ≤ 2 and 3 ≤ 3 but not 3 ≤ 2 Computer Security: Art and Science, 2 nd Edition Version 1.0 Slide A-4

Relation Properties • Reflexive • For all a Î S , aRa • On I , ≤ is reflexive as 1 ≤ 1, 2 ≤ 2, 3 ≤ 3 • Antisymmetric • For all a , b Î S , aRb Ù bRa Þ a = b • On I , ≤ is antisymmetric as 1 ≤ x and x ≤ 1 means x = 1 • Transitive • For all a , b , c Î S , aRb Ù bRc Þ aRc • On I , ≤ is transitive as 1 ≤ 2 and 2 ≤ 3 means 1 ≤ 3 Computer Security: Art and Science, 2 nd Edition Version 1.0 Slide A-5

Example • ℂ set of complex numbers • a Î ℂ Þ a = a R + a I i , where a R , a I integers • a ≤ C b if, and only if, a R ≤ b R and a I ≤ b I • a ≤ C b is reflexive, antisymmetric, transitive • As ≤ is over integers, and a R , a I are integers Computer Security: Art and Science, 2 nd Edition Version 1.0 Slide A-6

Partial Ordering • Relation R orders some members of set S • If all ordered, it ’ s a total ordering • Example • ≤ on integers is total ordering • ≤ ℂ is partial ordering on ℂ • Neither 3+5 i ≤ ℂ 4+2 i nor 4+2 i ≤ ℂ 3+5 i holds Computer Security: Art and Science, 2 nd Edition Version 1.0 Slide A-7

Upper Bounds • For a , b Î S , if u in S with aRu , bRu exists, then u is an upper bound • A least upper bound if there is no t Î S such that aRt , bRt , and tRu • Example • For 1 + 5 i , 2 + 4 i Î ℂ • Some upper bounds are 2 + 5 i , 3 + 8 i , and 9 + 100 i • Least upper bound is 2 + 5 i Computer Security: Art and Science, 2 nd Edition Version 1.0 Slide A-8

Lower Bounds • For a , b Î S , if l in S with lRa , lRb exists, then l is a lower bound • A greatest lower bound if there is no t Î S such that tRa , tRb , and lRt • Example • For 1 + 5 i , 2 + 4 i Î ℂ • Some lower bounds are 0, –1 + 2 i , 1 + 1 i , and 1+4 i • Greatest lower bound is 1 + 4 i Computer Security: Art and Science, 2 nd Edition Version 1.0 Slide A-9

Lattices • Set S , relation R • R is reflexive, antisymmetric, transitive on elements of S • For every s , t Î S , there exists a greatest lower bound under R • For every s , t Î S , there exists a least upper bound under R Computer Security: Art and Science, 2 nd Edition Version 1.0 Slide A-10

Example • S = { 0, 1, 2 }; R = ≤ is a lattice • R is clearly reflexive, antisymmetric, transitive on elements of S • Least upper bound of any two elements of S is the greater of the elements • Greatest lower bound of any two elements of S is the lesser of the elements Computer Security: Art and Science, 2 nd Edition Version 1.0 Slide A-11

Picture 2 1 0 Arrows represent ≤; this forms a total ordering Computer Security: Art and Science, 2 nd Edition Version 1.0 Slide A-12

Example • ℂ , ≤ ℂ form a lattice • ≤ ℂ is reflexive, antisymmetric, and transitive • Shown earlier • Least upper bound for a and b : • c R = max( a R , b R ), c I = max( a I , b I ); then c = c R + c I i • Greatest lower bound for a and b : • c R = min( a R , b R ), c I = min( a I , b I ); then c = c R + c I i Computer Security: Art and Science, 2 nd Edition Version 1.0 Slide A-13

Picture 2+5 i 1+5 i 2+4 i 1+4 i Arrows represent ≤ ℂ Computer Security: Art and Science, 2 nd Edition Version 1.0 Slide A-14